If exactly $245 is to be spent on ordering Pony 11-inch softballs, costing $2.75 each, and Junior 12-inch softballs, costing $3.25 each, how many softballs in total are to be ordered to minimize the difference between the numbers of each type of softball? (Note: the answer is NOT 80 softballs!)

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If exactly $245 is to be spent on ordering Pony 11-inch softballs, costing $2.75 each, and Junior 12-inch softballs, costing $3.25 each, how many softballs in total are to be ordered to minimize the difference between the numbers of each type of softball? (Note: the answer is NOT 80 softballs!)

Let P be the number of Pony 11-inch softballs @  $2.75 each

Let J be the number of Junior 12-inch softballs @  $3.25 each

Then our single Diophantine equation is:

2.75P + 3.25J = 245, i.e.

275P + 325J = 24500, which simplifies to

11P + 13J = 980

 

Now,

              13 = 1x11 + 2

              11 = 5x2 + 1

Rearranging the above equations, in reverse,

1 = 1x11 – 5x2

1 = 1x11 – 5(13 – 1x11)

1 = 6x11 – 5x13      now multiply by 980, giving

980 = 5880x11 – 4900x13

Equating the above to 980 = 11P + 13J gives us one solution, viz.

P = 5880

J = -4900

The general solution is,

P = 5880 – 13t

J = -4900 + 11t,

where the 13t and 11t use the coefficients of J and P, respectively, from the original Diophantine equation.

Both P and J must be positive values. Setting t = 446 gives us,

P = 82 – 13t

J = 6 + 11t,

and 0 <= t <= 6.

 

Let Δ = |P – J|

Δ = |82 – 13t – 6 – 11t|

Δ = |76 – 24t|

By observation we see that Δ is a minimum for t = 3. (Δ = |76 – 72| = 4)

At t = 3

P = 82 – 39 = 43

J = 6 + 33 = 39

P + J = 82

Answer: 82 softballs in total were ordered. 43 @ Pony 11-inch, and 39 @ Junior 12-inch softballs

by Level 11 User (81.5k points)
Spot on! Thanks for your answer.
I always like doing questions on Diophantine equations :)

Yes, they intrigue me, too.

I adopted a simpler approach based on the answer to the related question, where J=50, P=30. 

Using the relationship J=(245-2.75P)/3.25, and the fact that the two prices are 2.75=11/4 and 3.25=13/4, involving the integers 11 and 13, and a common denominator, I worked in steps of 13 from P=30 and steps of 11 from J=50 and came up withe following Diophantine pairs (P,J)=(4,72), (17,61), (30,50), (43,39), (56,28), (69,17), (82,6), giving the numbers difference: 68, 44, 20, 4, 28, 52 and 76. The fourth pair gives the minimum difference, making a total of 82 softballs.

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